Adding Edges to Increase the Chromatic Number of a Graph. (31st March 2016) - KOSTOCHKA, ALEXANDR; NEŠETŘIL, JAROSLAV
Abstract : If n ⩾ k + 1 and G is a connected n -vertex graph, then one can add k2 edges to G so that the resulting graph contains the complete graph K k +1 . This yields that for any connected graph G with at least k + 1 vertices, one can add k2 edges to G so that the resulting graph has chromatic number > k . A long time ago, Bollobás suggested that for every k ⩾ 3 there exists a k -chromatic graph Gk such that after adding to it any k2 − 1 edges, the chromatic number of the resulting graph is still k . In this note we prove this conjecture.